How does GIFT NIFTY operate?

Theoretically, a derivative is a contract on the underlying, whose price (or premium in case of options) should fluctuate based on how the underlying moves, and especially so with futures contracts which don’t have Greeks.

So why is it that GIFT NIFTY contract can be open when NIFTY50 index constituents are not trading in cash market? What is the logic here? And what decides it’s movement if it’s not the underlying index at that point in time?

Is futures price a mathematical derivative of the index price? It is whatever people are willing to pay for the futures contract right. Not sure how it works.

There is no defined logic, it just trades based on demand and supply and that demand and supply comes based on perceptions/opinions and views. For ex in most cases if US markets are high gift tends to go up, if asian markets are up gift tends to be in green.
Few times new market data ( Inflation numbers,GDP etc) comes out during off market hours and in that case based on output gift discounts it before normal Nifty open.
Also there are many cases where we can notice Nifty may not open up to the price what GIft is trading at because gift nifty is just trading based on opinions with out any underlying so one should not worry much about where Gift nifty is trading at during off market hours and only should be used to just get a broad opinion.

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I understand that a derivative contract can influence the underlying. But how can a contract itself trade if the underlying is not trading? On what basis is the contract trading then?

Hi, speculation on anticipated future price movements and overnight risk hedging by large entities.

As mentioned above it is traded on the basis of expectations with discounting new information available , if any.